We discuss some problems and conjectures in higher dimensional operator
theory. These all have something to do with the basic problem of developing
an effective Fredholm theory of d-contractions, and completing the index
theorem that was partially established in Theorem B of [Arv02 ], following up
on [Arv00 ]. My opinion is that good progress on any one of these problems
will be a significant advance, if not a breakthrough.